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Theorem ixxss1 10890
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxss1.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
ixxss1.3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
Assertion
Ref Expression
ixxss1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, O   
w, B, x, y, z    w, P    x, R, y, z    x, S, y, z    x, T, y, z    w, W
Allowed substitution hints:    P( x, y, z)    R( w)    S( w)    T( w)    O( x, y, z)    W( x, y, z)

Proof of Theorem ixxss1
StepHypRef Expression
1 ixxss1.2 . . . . . . . 8  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
21elixx3g 10885 . . . . . . 7  |-  ( w  e.  ( B P C )  <->  ( ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  /\  ( B T w  /\  w S C ) ) )
32simplbi 447 . . . . . 6  |-  ( w  e.  ( B P C )  ->  ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* ) )
43adantl 453 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B  e.  RR*  /\  C  e. 
RR*  /\  w  e.  RR* ) )
54simp3d 971 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  RR* )
6 simplr 732 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A W B )
72simprbi 451 . . . . . . 7  |-  ( w  e.  ( B P C )  ->  ( B T w  /\  w S C ) )
87adantl 453 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B T w  /\  w S C ) )
98simpld 446 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B T w )
10 simpll 731 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A  e.  RR* )
114simp1d 969 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
12 ixxss1.3 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
1310, 11, 5, 12syl3anc 1184 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( ( A W B  /\  B T w )  ->  A R w ) )
146, 9, 13mp2and 661 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A R w )
158simprd 450 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w S C )
164simp2d 970 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  C  e.  RR* )
17 ixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 10881 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1910, 16, 18syl2anc 643 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( w  e.  ( A O C )  <->  ( w  e. 
RR*  /\  A R w  /\  w S C ) ) )
205, 14, 15, 19mpbir3and 1137 . . 3  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  ( A O C ) )
2120ex 424 . 2  |-  ( ( A  e.  RR*  /\  A W B )  ->  (
w  e.  ( B P C )  ->  w  e.  ( A O C ) ) )
2221ssrdv 3314 1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2670    C_ wss 3280   class class class wbr 4172  (class class class)co 6040    e. cmpt2 6042   RR*cxr 9075
This theorem is referenced by:  iooss1  10907  limsupgord  12221  pnfnei  17238  dvfsumrlimge0  19867  dvfsumrlim2  19869  tanord1  20392  rlimcnp  20757  rlimcnp2  20758  dchrisum0lem2a  21164  pntleml  21258  pnt  21261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-xr 9080
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